I'm a little confused with using derivatives. I understand that derivatives respond to question
how result of function changes when input of function changes
However I do not understand why we calculate derivative of $f(x) = x^2$ instead of calculating $f(2)$ and $f(2.001)$ and subtracting these two values of $f(2.001)$ from $f(2.0)$
Could someone explain that?
Indeed, these are almost the same thing. The calculation $$\frac{f(2.001)-f(2)}{0.001}$$ is an approximation to the exact rate of change at $x=2$. Note that the rate of change isn't just $f(2.001)-f(2)$; that's just the change in the value of the function. You need to divide by the amount of change in $x$ to get the rate of change; this is often expressed $\Delta y / \Delta x$.
For a much better approximation, you could calculate $$\frac{f(2.00000001)-f(2)}{0.00000001}$$
Ultimately, the derivative is defined to be the limit of such expressions: $$f'(2) := \lim_{h\to0}\frac{f(2+h)-f(2)}{h}$$
Finally, note that in addition to being exact (and not just an approximation), it is much simpler to calculate $f'(x)$, since in this example it is just $2x$. That being said, in the "real world", we don't always have nice functions like $x^2$, but rather only numerical data. In that case, you can approximate the derivative as above.
